A coupled double-quantum-well resonant tunneling structure in the presence of electron-phonon interaction is investigated carefully using the nonequilibrium Green's functions method. The tunneling current formula through this system under a de voltage has been derived. Two special cases are considered which show how LO phonon-emission-assisted and LO phonon-absorption-assisted tunneling come into effect. In some cases, the LO phonon-emission-assisted tunneling is suppressed due to the Pauli exclusion principle, while the LO phonon-absorption-assisted tunneling is much more prominent when elastic resonant tunneling is impossible.